$a^n+ b^n$ is not a prime number when $n$ is not a power of $2$.

Question

Suppose $a, b$ and $n$ are positive integers, $a+b>2$, and $n$ is not a power of $2$. Prove that $a^n+ b^n$ is not a prime number.

Answer 1

You can't "assume $n$ to be odd", but you can say that it has an odd factor $> 1$.

Answer 2

You can not assume it's odd, but you can assume it contains an odd factor.

Consider $n=2^k\cdot x$ where $x$ is odd.

$$a^n+b^n=a^{2^k\cdot x}+b^{2^k\cdot x}=(a^{2^k}+b^{2^k})(a^{2^k\cdot(x-1)}-a^{2^k\cdot(x-2)}b^{2^k}+...-b^{2^k\cdot(x-2)}a^{2^k}+b^{2^k\cdot(x-1)})$$

Since both brackets are greater than $1$, $a^n+b^n$ is composite.